Question: Find the largest solution to \[\lfloor x \rfloor = 5 + 100 \{ x \},\]where $\{x\} = x - \lfloor x \rfloor.$
Answer: Since $0 \le \{x\} < 1,$ we have $0 \le 100 \{x\} < 100,$ so $5 \le 5 + 100 \{x\} < 105.$ Thus, \[5 \le \lfloor x\rfloor < 105.\]Since $\lfloor x\rfloor$ is an integer, the possible values of $\lfloor x\rfloor$ are $5, 6, \dots, 104.$ For each of these values of $\lfloor x\rfloor,$ we get a corresponding value \[\{x\} = \frac{\lfloor x\rfloor - 5}{100} = 0.01 \lfloor x \rfloor - 0.05,\]and then we have \[x = \lfloor x\rfloor + \{x\} = 1.01 \lfloor x \rfloor - 0.05.\]To maximize $x,$ we choose $\lfloor x \rfloor = 104,$ giving \[x = 1.01 \cdot 104 - 0.05 = \boxed{104.99}.\]